A309797 Lexicographically earliest sequence of positive integers such that for any n > 0 there are no more than a(n) numbers k > 0 such that a(n + k) = a(n + 2*k).

1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 3, 2, 2, 2, 4, 2, 4, 5, 4, 1, 1, 1, 1, 3, 3, 3, 1, 6, 6, 6, 7, 5, 3, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 7, 5, 7, 5, 8, 5, 8, 8, 8, 8, 3, 1, 1, 1, 1, 9, 9, 6, 1, 1, 1, 1, 10, 3, 3, 3, 6, 6, 6, 6, 6, 7, 6, 3, 9, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Rémy Sigrist, Nov 11 2019

Keywords

• nonn
• look

Comments

The sequence is well defined as we can always extend the sequence with a number that has not yet appeared.

The number 1 appears infinitely many times in the sequence:

- by contradiction: suppose that m is the index of the last occurrence of 1 in the sequence,

- there is no n > 0 such that n + k = m and n + 2*k = 2*m (with k > 0),

- so we can choose a(2*m) = 1, QED.

This sequence has connections with A003602:

- here we have up to a(n) numbers k such that a(n+k) = a(n+2*k), there we have no such numbers,

- for any v >= 0, let f_v be the lexicographically earliest sequence of positive integers such that there are no more than v numbers k such that f_v(n + k) = f_v(n + 2*k),

- then f_v corresponds to A003602 where all but the first term have been repeated 2*v+1 times.

Examples

			The first terms, alongside the corresponding k's, are:
  n   a(n)  k's
  --  ----  ----------
   1     1  {1}
   2     1  {1}
   3     1  {2}
   4     1  {1}
   5     2  {1, 3}
   6     2  {2, 14}
   7     2  {1, 2}
   8     1  {1}
   9     1  {1}
  10     1  {4}
  11     1  {1}
  12     3  {2, 3, 68}
  13     3  {1, 4, 22}
  14     2  {1, 2}
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A309797
• Rémy Sigrist, Scatterplot of the first 500000 terms

Crossrefs

Cf. A003602, A329268 (positions of 1).

Programs

PARI

See Links section.

Formula

a(n) >= #{ k>0 such that a(n+k) = a(n+2*k) }.